Support of Noise Standard Deviations Specified For Some Points

For a common approximation model construction problem, the user has only a set of points and corresponding target function values. Using this training data, an approximation model is constructed. However, in some cases, the user can also provide variances of the target function values (output variances) for some points from the sample. GT Approx can handle such kind of information, as output variances are used to construct a approximation model of noisy functions.

This example demonstrates the usage of partly provided output variances with a model example. We compare approximation models constructed with and without the use of output variances for a model function.

Problem Statement and Solution

Here a one-dimensional model example is considered. The objective function is shown in the picture below.

Note that in this example only highly noisy values of the target function are available.

• To build an approximation model, we need a training data, which is then passed to GT Approx builder.
• The true function is highly noisy. So it is deliberate to pass the initial function values to GT Approx. Below we present an approach to proceed initial data, so we can pass it to GT Approx builder.
• GT DoE is used to generate training points.
• Then we evaluate the target function for each training point several times.
• We estimate mean value and variance for each point with casual engineering routines.
• We assume that output variances for some points are not available. To make the demonstration more efficient, output variances for some points are not considered as if they were not known at all, which makes the problem solution more complex.

Now we are ready to construct approximation models with and without the use of output variances. We run GT Approx in both modes: first, only the points and corresponding target function values are used. At the second run, we use points, corresponding target function values and output variances for some of this points. Let’s compare the two resulting approximation models.

The most reliable way to compare different approximation models is to calculate errors of approximation for a separate test data, which consists of target function values in new points. This method is applied.

Results

• Using of output variances during approximation model construction decreased errors from RRMS (Relative Root Mean Square Error) value 0.3650 to RRMS value 0.2127 by about 40%

The picture below illustrates the obtained approximation models.

One can see that Accuracy Evaluation (uncertainty in the approximation model prediction) is more accurate if partly specified output variances are used. It proves that using partly specified output variances is really efficient.